Two asymptotic expansions for gamma function developed by Windschitl’s formula

In this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have$$\begin{array}{} \displaystyle {\it \Gamma} \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1} {\frac{a_{n}}{x^{2n-1}}}+R_{n}\left( x\right) \right) \end{array}$$with$$\begin{array}{} \displaystyle \left\vert R_{n}\left( x\right) \right\vert \leq \frac{\left\vert B_{2n}\right\vert }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{array}$$for all x > 0, where an has a closed-form expression, B2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.